By Mickelsson J.
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Extra info for Advanced quantum mechanics
1 to give meaning to a sum over paths for an action which is not quadratic. Given its utility in the study of the relativistic particle in the next section, we will ﬁrst develop this approach in the non-relativistic context. In classical mechanics the usual form of the action A[x(t)] ≡ A[xα (t)] is taken to be a time integral [in the range (t1 , t2 )] of the Lagrangian L = T (xα , x˙ α )−V (xα ). Demanding δA = 0 with δxα = 0 at the end points leads to the equations of motion determining the trajectory xα (t) satisfying the boundary conditions.
This is because, in the relativistic case, we no longer have a |x1 = |t1 , x1 which is an eigenket of some sensible posiˆ 1 ) with eigenvalue x1 . tion operator x(t We do have such an eigenket in the NRQM. Interpreting G(x2; x1) in Terms of a Field All this is vaguely disturbing to someone who is accustomed to sensible physical evolution which proceeds monotonously forward in time from t1 < t2 < t3 .... 32 We have already seen that G(x2 ; x1 ) comes from summing over paths which include the likes of the one in Figs.
11), except for two crucial diﬀerences. First, the non-relativistic amplitude propagates modes with positive energy ωp forward in time and we could take it to be zero for t < 0. The relativistic amplitude, on the other hand, has |t| leading to propagation both forward and backward in time. It propagates modes with energy ωp forward in time, but also propagates modes with negative energy −ωp backwards in time, because ωp |t| = (−ωp )t for t < 0. 23 What about the c → ∞ limit when we expect to recover the nonrelativistic form of G(x2 ; x1 ) from the relativistic expression?
Advanced quantum mechanics by Mickelsson J.